We study a learning-augmented variant of the secretary problem, recently introduced by Fujii and Yoshida (2023), in which the decision-maker has access to machine-learned predictions of candidate values. The central challenge is to balance consistency and robustness: when predictions are accurate, the algorithm should select a near-optimal secretary, while under inaccurate predictions it should still guarantee a bounded competitive ratio. We consider both the classical Random Order Secretary Problem (ROSP), where candidates arrive in a uniformly random order, and a more natural learning-augmented model in which the decision-maker may choose the arrival order based on predicted values. We call this model the Chosen Order Secretary Problem (COSP), capturing scenarios such as interview schedules set in advance. We propose a new randomized algorithm applicable to both ROSP and COSP. Our method switches from fully trusting predictions to a threshold-based rule once a large prediction deviation is detected. Let $ε\in [0,1]$ denote the maximum multiplicative prediction error. For ROSP, our algorithm achieves a competitive ratio of $\max\{0.221, (1-ε)/(1+ε)\}$, improving upon the prior bound of $\max\{0.215, (1-ε)/(1+ε)\}$. For COSP, we achieve $\max\{0.262, (1-ε)/(1+ε)\}$, surpassing the $0.25$ worst-case bound for prior approaches and moving closer to the classical secretary benchmark of $1/e \approx 0.368$. These results highlight the benefit of combining predictions with arrival-order control in online decision-making.

翻译：我们研究了由 Fujii 和 Yoshida (2023) 最近引入的一种学习增强型秘书问题变体，其中决策者可以访问机器学习预测的候选人价值。核心挑战在于平衡一致性与鲁棒性：当预测准确时，算法应选择接近最优的秘书；而在预测不准确的情况下，算法仍需保证一个有界的竞争比。我们既考虑了经典的随机顺序秘书问题（ROSP），即候选人以均匀随机顺序到达，也考虑了一个更自然的学习增强模型，其中决策者可以根据预测价值选择到达顺序。我们将此模型称为选择顺序秘书问题（COSP），它捕捉了诸如预先设定面试日程等场景。我们提出了一种适用于 ROSP 和 COSP 的新型随机化算法。我们的方法在检测到较大的预测偏差时，会从完全信任预测切换到基于阈值的规则。令 $ε\in [0,1]$ 表示最大乘性预测误差。对于 ROSP，我们的算法实现了 $\max\{0.221, (1-ε)/(1+ε)\}$ 的竞争比，优于先前的 $\max\{0.215, (1-ε)/(1+ε)\}$ 界限。对于 COSP，我们实现了 $\max\{0.262, (1-ε)/(1+ε)\}$ 的竞争比，超越了先前方法 $0.25$ 的最坏情况界限，并更接近经典秘书问题的基准 $1/e \approx 0.368$。这些结果突显了在在线决策中将预测与到达顺序控制相结合的优势。